User blog:Ikosarakt1/Extension of Chris Bird's array notation.
In this blog post, I shall write about my extension of Chris Bird's array notation. About separators with level below \(\varepsilon_{\Omega+1}\) (i.e. Bachmann-Howard ordinal) and the gist of array notation you can read here. Define Bird's hierarchy: \(B_{\alpha}(n) = \lbrace n,n X 2 \rbrace\), where \(\alpha\) specifies a separator and vice versa. Notice that Bird's hierarchy grows faster even than fast-growing, for example: \(f_{\varepsilon_0+1}(n) \approx \lbrace n,n,2 \backslash 2 2 \rbrace\) \(B_{\varepsilon_0+1}(n) \approx \lbrace n,n \backslash 2 2 \rbrace\), since \(\backslash 2\) has level \(\varepsilon_0+1\) In 7 part of Bird's stuff, he defines "generalized" backslash separator: \(\backslash _n\). There are good, but what about turning n into array itself? I define: \(\lbrace n,n \backslash _{[1,2} 2] 2 \rbrace = H(n)\), this function was defined by Bird at the end of paper. Now I show list of separators beyond this (for the time being, without complete set of formal rules). Let \(\rightarrow\) is shorthand for "has level". \(\backslash _{[1,2} 2] \rightarrow \vartheta(\varepsilon_{\Omega+1})\) \(\backslash _{[1,2} 2] \rightarrow \vartheta(\varepsilon_{\Omega+1})+1\) \(\backslash _{[1,2} 2] \rightarrow \vartheta(\varepsilon_{\Omega+1})+2\) \(\backslash _{[1,2} 2] \rightarrow \vartheta(\varepsilon_{\Omega+1})+\omega\) \([2 2 \backslash _{1,2} 2] \rightarrow \vartheta(\varepsilon_{\Omega+1})+\omega^2\) \([1 \backslash 2 2 \backslash _{1,2} 2] \rightarrow \vartheta(\varepsilon_{\Omega+1})+\varepsilon_0\) \([2 \backslash 2 2 \backslash _{1,2} 2] \rightarrow \vartheta(\varepsilon_{\Omega+1})^2\) \(\backslash 2 \backslash _{[1,2} 2] \rightarrow \varepsilon_{\vartheta(\varepsilon_{\Omega+1})+1}\) \(\backslash 1 \backslash 2 \backslash _{[1,2} 2] \rightarrow \zeta_{\vartheta(\varepsilon_{\Omega+1})+1}\) \(\backslash 1 \backslash 1 \backslash 2 \backslash _{[1,2} 2] \rightarrow \eta_{\vartheta(\varepsilon_{\Omega+1})+1}\) \(\backslash _{[1,2} 3] \rightarrow \vartheta(\varepsilon_{\Omega+1},1)\) \(\backslash _{[1,2} 4] \rightarrow \vartheta(\varepsilon_{\Omega+1},2)\) \(\backslash _{[1,2} 1,2] \rightarrow \vartheta(\varepsilon_{\Omega+1},\omega)\) \(\backslash _{[1,2} 1 \backslash _{[1,2} 2] 2] \rightarrow \vartheta(\varepsilon_{\Omega+1},\vartheta(\varepsilon_{\Omega+1}))\) \(\backslash _{[1,2} 1 \backslash 2] \rightarrow \vartheta(\varepsilon_{\Omega+1}+1)\) \(\backslash _{[1,2} 1 \backslash 1 \backslash 2] \rightarrow \vartheta(\varepsilon_{\Omega+1}+2)\) \(\backslash _{[1,2} 1 \backslash _{1,2} 2] \rightarrow \vartheta(\varepsilon_{\Omega+1}+\vartheta(\varepsilon_{\Omega+1}))\) \([2 \backslash _{[2,2} 2] 2] \rightarrow \vartheta(\varepsilon_{\Omega 2})\) \([3 \backslash _{[2,2} 2] 2] \rightarrow \vartheta(\varepsilon_{\Omega 3})\) \([1,2 \backslash _{[2,2} 2] 2] \rightarrow \vartheta(\varepsilon_{\Omega \omega})\) \([1 \backslash _{[2,2} 3] 2] \rightarrow \vartheta(\varepsilon_{\Omega^2})\) \([1 \backslash _{[2,2} 4] 2] \rightarrow \vartheta(\varepsilon_{\Omega^3})\) \([1 \backslash _{[2,2} 1,2] 2] \rightarrow \vartheta(\varepsilon_{\Omega^\omega})\) \([1 \backslash _{[2,2} 1 \backslash _{2,2} 2] 2] \rightarrow \vartheta(\varepsilon_{\Omega^\Omega})\) \([1 [2 \backslash _{[3,2} 2] 2] 2] \rightarrow \vartheta(\varepsilon_{\Omega^{\Omega^\omega}})\) \([1 [1 \backslash _{[3,2} 3] 2] 2] \rightarrow \vartheta(\varepsilon_{\Omega^{\Omega^\Omega}})\) \([1 [1 \backslash _{[3,2} 1,2] 2] 2] \rightarrow \vartheta(\varepsilon_{\Omega^{\Omega^{\Omega^{\Omega^\omega}}}})\) \([1 [1 \backslash _{[3,2} 1 \backslash _{3,2} 2] 2] 2] \rightarrow \vartheta(\varepsilon_{\Omega^{\Omega^{\Omega^{\Omega^\Omega}}}})\) \([1 [1 [2 \backslash _{[4,2} 2] 2] 2] 2] \rightarrow \vartheta(\varepsilon_{\Omega^{\Omega^{\Omega^{\Omega^{\Omega^w}}}}})\) \(\backslash _{[1,3} 2] \rightarrow \vartheta(\varepsilon_{\varepsilon_{\Omega+1}})\) \(\backslash _{[1,4} 2] \rightarrow \vartheta(\varepsilon_{\varepsilon_{\Omega+1}})\) \(\backslash _{[1,1,2} 2] \rightarrow \vartheta(\zeta_{\Omega+1})\) \(\backslash _{[1,1,1,2} 2] \rightarrow \vartheta(\eta_{\Omega+1})\) \(\backslash _{[1 [2 2]} 2] \rightarrow \vartheta(\vartheta(\omega),{\Omega+1})\) Now we can perform the following transformation: \(X_2 = \backslash _{[X} 2]\). It allows us to write next separators more compactly: \([3 2]_2 \rightarrow \vartheta(\vartheta(\omega^2),{\Omega+1})\) \([4 2]_2 \rightarrow \vartheta(\vartheta(\omega^3),{\Omega+1})\) \([1,2 2]_2 \rightarrow \vartheta(\vartheta(\omega^\omega),{\Omega+1})\) \([1 [2 2] 2]_2 \rightarrow \vartheta(\vartheta(\omega^{\omega^\omega}),{\Omega+1})\) \([1 \neg 3 2]_2 \rightarrow \vartheta(\Omega_2)\), where \(\Omega_2\) is a second uncountable ordinal. Generally, \(X_{n \#} = \backslash _{[X_{n-1 \#}} 2]_{n-1 \#} \). \([1 \neg 3 2]_n \rightarrow \vartheta(\Omega_n)\), where \(\Omega_n\) is a n-th uncountable ordinal. \([1 \neg 3 2]_{1,2} \rightarrow \vartheta(\Omega_{\omega})\) The last entry is Takeuti-Feferman-Buchholz ordinal. Growth rate of its finite form is comparable to the lower bound for Friedman's SCG(n). \([1 \neg 3 2]_{1,1,2} \rightarrow \vartheta(\Omega_{\omega^2})\) \([1 \neg 3 2]_{1 2 2} \rightarrow \vartheta(\Omega_{\omega^\omega})\) \([1 \neg 3 2]_{1 3 2} \rightarrow \vartheta(\Omega_{\omega^{\omega^2}})\) \([1 \neg 3 2]_{1 1,2 2} \rightarrow \vartheta(\Omega_{\omega^{\omega^\omega}})\) \([1 \neg 3 2]_{[1 \neg 3 2]} \rightarrow \vartheta(\Omega_{\Omega})\) \([1 \neg 3 2]_{[1 \neg 3 2]_{\cdots [1 \neg 3 2]}} \rightarrow \vartheta(\Omega_{\Omega_{\cdots \Omega}})\) Of course, there are need more rules to define it explicitly, but it potentially can match \(\vartheta(\Omega_{\Omega_{\Omega_{\cdots \Omega}}})\), with arbitrary number of \(\Omega\)'s. Category:Blog posts